To determine which statement is false among the given options, we need to analyze the provided details about the set of numbers:
1. Mode = 90: This means that 90 is the most frequently occurring number in the set.
2. Mean = 80: The arithmetic mean of the five numbers is 80.
3. Total numbers in the set = 5: The set consists of exactly 5 numbers.
First, calculate the sum of the five numbers:
[tex]\[
\text{Sum of the numbers} = \text{Mean} \times \text{Total numbers} = 80 \times 5 = 400
\][/tex]
This verifies statement (D):
[tex]\[
\text{(D) The five numbers must have a sum of 400.}
\][/tex]
This statement is true.
Now let’s analyze the other statements:
(A) The number 90 appears two, three, or four times in the set:
Since 90 is the mode, it must appear more frequently than any other number in the set. If it appeared only once, it wouldn’t be the mode; hence, it must appear at least twice. The upper limit of its appearance would be four times, as appearing five times means all numbers would be 90, leading to a mean of 90 instead of 80.
This statement is true.
(B) The number 240 cannot appear in the set:
The number 240 is extremely high compared to the other numbers, considering the mean is 80. Including such a high number would require exceedingly low values for the other numbers to maintain the mean of 80, which we would verify through theoretical calculations. But since the sum is 400, and including 240 would necessitate that the remaining four numbers sum to 160, an impossibility with positive whole numbers and the given mode constraints.
Hence, this statement is true.
(C) The number 80 must appear exactly once in the set:
While the mean is 80, this does not necessarily mean that the number 80 must appear in the set. For instance, with the mode being 90, other configurations of numbers could still balance out to a mean of 80 without including the number 80 itself.
Thus, this statement is false.
In summary:
- (A) is true.
- (B) is true.
- (C) is false.
- (D) is true.
The false statement is (C): "The number 80 must appear exactly once in the set."
